Existence of tangent lines to Carnot-Carathéodory geodesics

We prove that length minimizing curves in Carnot-Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line. This is the first regularity result for length minimizers that holds with no assumption on either the space (e.g., its rank, step, or analyticity) or the curve.